Definition:Bounded Above Mapping/Real-Valued

This page is about Bounded Above in the context of Real-Valued Function. For other uses, see Bounded Above.

Definition

Let $f: S \to \R$ be a real-valued function.

$f$ is bounded above on $S$ by the upper bound $H$ if and only if:

$\forall x \in S: \map f x \le H$

That is, if and only if the set $\set {\map f x: x \in S}$ is bounded above in $\R$ by $H$.

Unbounded Above

Let $f: S \to \R$ be a real-valued function.

Then $f$ is unbounded above on $S$ if and only if it is not bounded above on $S$:

$\neg \exists H \in \R: \forall x \in S: \map f x \le H$

Also see

• Definition:Upper Bound of Real-Valued Function

• Definition:Bounded Below Real-Valued Function
• Definition:Lower Bound of Real-Valued Function

• Definition:Bounded Real-Valued Function

• Results about bounded above real-valued functions can be found here.

Sources

• 1947: James M. Hyslop: Infinite Series (3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 3$: Bounds of a Function
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 7.13$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bound: 1. (of a function)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bound: 1. (of a function)